Step | Hyp | Ref
| Expression |
1 | | gsumhashmul.f |
. . . . . . 7
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
2 | | suppssdm 7844 |
. . . . . . . 8
⊢ (𝐹 supp 0 ) ⊆ dom 𝐹 |
3 | 2, 1 | fssdm 6508 |
. . . . . . 7
⊢ (𝜑 → (𝐹 supp 0 ) ⊆ 𝐴) |
4 | 1, 3 | feqresmpt 6715 |
. . . . . 6
⊢ (𝜑 → (𝐹 ↾ (𝐹 supp 0 )) = (𝑥 ∈ (𝐹 supp 0 ) ↦ (𝐹‘𝑥))) |
5 | 4 | oveq2d 7159 |
. . . . 5
⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐹 supp 0 ))) = (𝐺 Σg (𝑥 ∈ (𝐹 supp 0 ) ↦ (𝐹‘𝑥)))) |
6 | | gsumhashmul.b |
. . . . . 6
⊢ 𝐵 = (Base‘𝐺) |
7 | | gsumhashmul.z |
. . . . . 6
⊢ 0 =
(0g‘𝐺) |
8 | | gsumhashmul.g |
. . . . . 6
⊢ (𝜑 → 𝐺 ∈ CMnd) |
9 | | gsumhashmul.1 |
. . . . . . . 8
⊢ (𝜑 → 𝐹 finSupp 0 ) |
10 | | relfsupp 8853 |
. . . . . . . . 9
⊢ Rel
finSupp |
11 | 10 | brrelex1i 5570 |
. . . . . . . 8
⊢ (𝐹 finSupp 0 → 𝐹 ∈ V) |
12 | 9, 11 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐹 ∈ V) |
13 | 1 | ffnd 6492 |
. . . . . . 7
⊢ (𝜑 → 𝐹 Fn 𝐴) |
14 | 12, 13 | fndmexd 7609 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ V) |
15 | | ssidd 3911 |
. . . . . 6
⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )) |
16 | 6, 7, 8, 14, 1, 15, 9 | gsumres 19086 |
. . . . 5
⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐹 supp 0 ))) = (𝐺 Σg 𝐹)) |
17 | | nfcv 2917 |
. . . . . 6
⊢
Ⅎ𝑥(𝐹‘(1st ‘𝑧)) |
18 | | fveq2 6651 |
. . . . . 6
⊢ (𝑥 = (1st ‘𝑧) → (𝐹‘𝑥) = (𝐹‘(1st ‘𝑧))) |
19 | 9 | fsuppimpd 8858 |
. . . . . 6
⊢ (𝜑 → (𝐹 supp 0 ) ∈
Fin) |
20 | | ssidd 3911 |
. . . . . 6
⊢ (𝜑 → 𝐵 ⊆ 𝐵) |
21 | 1 | adantr 485 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → 𝐹:𝐴⟶𝐵) |
22 | 3 | sselda 3888 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → 𝑥 ∈ 𝐴) |
23 | 21, 22 | ffvelrnd 6836 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → (𝐹‘𝑥) ∈ 𝐵) |
24 | 1 | ffund 6495 |
. . . . . . . . 9
⊢ (𝜑 → Fun 𝐹) |
25 | | funrel 6345 |
. . . . . . . . 9
⊢ (Fun
𝐹 → Rel 𝐹) |
26 | | reldif 5650 |
. . . . . . . . 9
⊢ (Rel
𝐹 → Rel (𝐹 ∖ (V × { 0
}))) |
27 | 24, 25, 26 | 3syl 18 |
. . . . . . . 8
⊢ (𝜑 → Rel (𝐹 ∖ (V × { 0 }))) |
28 | | 1stdm 7736 |
. . . . . . . 8
⊢ ((Rel
(𝐹 ∖ (V × {
0 }))
∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) →
(1st ‘𝑧)
∈ dom (𝐹 ∖ (V
× { 0 }))) |
29 | 27, 28 | sylan 584 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) →
(1st ‘𝑧)
∈ dom (𝐹 ∖ (V
× { 0 }))) |
30 | 7 | fvexi 6665 |
. . . . . . . . . . . 12
⊢ 0 ∈
V |
31 | 30 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → 0 ∈ V) |
32 | | fressupp 30531 |
. . . . . . . . . . 11
⊢ ((Fun
𝐹 ∧ 𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 }))) |
33 | 24, 12, 31, 32 | syl3anc 1369 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 }))) |
34 | 33 | dmeqd 5738 |
. . . . . . . . 9
⊢ (𝜑 → dom (𝐹 ↾ (𝐹 supp 0 )) = dom (𝐹 ∖ (V × { 0
}))) |
35 | 2 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐹 supp 0 ) ⊆ dom 𝐹) |
36 | | ssdmres 5839 |
. . . . . . . . . 10
⊢ ((𝐹 supp 0 ) ⊆ dom 𝐹 ↔ dom (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 supp 0 )) |
37 | 35, 36 | sylib 221 |
. . . . . . . . 9
⊢ (𝜑 → dom (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 supp 0 )) |
38 | 34, 37 | eqtr3d 2796 |
. . . . . . . 8
⊢ (𝜑 → dom (𝐹 ∖ (V × { 0 })) = (𝐹 supp 0 )) |
39 | 38 | adantr 485 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → dom (𝐹 ∖ (V × { 0 })) = (𝐹 supp 0 )) |
40 | 29, 39 | eleqtrd 2853 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) →
(1st ‘𝑧)
∈ (𝐹 supp 0
)) |
41 | 24 | funresd 6371 |
. . . . . . . . . . 11
⊢ (𝜑 → Fun (𝐹 ↾ (𝐹 supp 0 ))) |
42 | 41 | adantr 485 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → Fun (𝐹 ↾ (𝐹 supp 0 ))) |
43 | 37 | eleq2d 2836 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 )) ↔ 𝑥 ∈ (𝐹 supp 0 ))) |
44 | 43 | biimpar 482 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → 𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) |
45 | | simpr 489 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → 𝑥 ∈ (𝐹 supp 0 )) |
46 | 45 | fvresd 6671 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹‘𝑥)) |
47 | | funopfvb 6702 |
. . . . . . . . . . 11
⊢ ((Fun
(𝐹 ↾ (𝐹 supp 0 )) ∧ 𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) → (((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹‘𝑥) ↔ 〈𝑥, (𝐹‘𝑥)〉 ∈ (𝐹 ↾ (𝐹 supp 0 )))) |
48 | 47 | biimpa 481 |
. . . . . . . . . 10
⊢ (((Fun
(𝐹 ↾ (𝐹 supp 0 )) ∧ 𝑥 ∈ dom (𝐹 ↾ (𝐹 supp 0 ))) ∧ ((𝐹 ↾ (𝐹 supp 0 ))‘𝑥) = (𝐹‘𝑥)) → 〈𝑥, (𝐹‘𝑥)〉 ∈ (𝐹 ↾ (𝐹 supp 0 ))) |
49 | 42, 44, 46, 48 | syl21anc 837 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → 〈𝑥, (𝐹‘𝑥)〉 ∈ (𝐹 ↾ (𝐹 supp 0 ))) |
50 | 33 | adantr 485 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 ∖ (V × { 0 }))) |
51 | 49, 50 | eleqtrd 2853 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → 〈𝑥, (𝐹‘𝑥)〉 ∈ (𝐹 ∖ (V × { 0 }))) |
52 | | eqeq2 2771 |
. . . . . . . . . . 11
⊢ (𝑣 = 〈𝑥, (𝐹‘𝑥)〉 → (𝑧 = 𝑣 ↔ 𝑧 = 〈𝑥, (𝐹‘𝑥)〉)) |
53 | 52 | bibi2d 347 |
. . . . . . . . . 10
⊢ (𝑣 = 〈𝑥, (𝐹‘𝑥)〉 → ((𝑥 = (1st ‘𝑧) ↔ 𝑧 = 𝑣) ↔ (𝑥 = (1st ‘𝑧) ↔ 𝑧 = 〈𝑥, (𝐹‘𝑥)〉))) |
54 | 53 | ralbidv 3124 |
. . . . . . . . 9
⊢ (𝑣 = 〈𝑥, (𝐹‘𝑥)〉 → (∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st ‘𝑧) ↔ 𝑧 = 𝑣) ↔ ∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st ‘𝑧) ↔ 𝑧 = 〈𝑥, (𝐹‘𝑥)〉))) |
55 | 54 | adantl 486 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑣 = 〈𝑥, (𝐹‘𝑥)〉) → (∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st ‘𝑧) ↔ 𝑧 = 𝑣) ↔ ∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st ‘𝑧) ↔ 𝑧 = 〈𝑥, (𝐹‘𝑥)〉))) |
56 | | fvexd 6666 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → (2nd
‘𝑧) ∈
V) |
57 | 27 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → Rel (𝐹 ∖ (V × { 0 }))) |
58 | | simplr 769 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) |
59 | | 1st2nd 7735 |
. . . . . . . . . . . . . . . . 17
⊢ ((Rel
(𝐹 ∖ (V × {
0 }))
∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) →
𝑧 = 〈(1st
‘𝑧), (2nd
‘𝑧)〉) |
60 | 57, 58, 59 | syl2anc 588 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝑧 = 〈(1st ‘𝑧), (2nd ‘𝑧)〉) |
61 | | opeq1 4754 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = (1st ‘𝑧) → 〈𝑥, (2nd ‘𝑧)〉 = 〈(1st
‘𝑧), (2nd
‘𝑧)〉) |
62 | 61 | adantl 486 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 〈𝑥, (2nd ‘𝑧)〉 = 〈(1st
‘𝑧), (2nd
‘𝑧)〉) |
63 | 60, 62 | eqtr4d 2797 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝑧 = 〈𝑥, (2nd ‘𝑧)〉) |
64 | | difssd 4034 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → (𝐹 ∖ (V × { 0 })) ⊆
𝐹) |
65 | 64 | sselda 3888 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧 ∈ 𝐹) |
66 | 65 | adantr 485 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝑧 ∈ 𝐹) |
67 | 63, 66 | eqeltrrd 2852 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 〈𝑥, (2nd ‘𝑧)〉 ∈ 𝐹) |
68 | 63, 67 | jca 516 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → (𝑧 = 〈𝑥, (2nd ‘𝑧)〉 ∧ 〈𝑥, (2nd ‘𝑧)〉 ∈ 𝐹)) |
69 | | opeq2 4756 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = (2nd ‘𝑧) → 〈𝑥, 𝑦〉 = 〈𝑥, (2nd ‘𝑧)〉) |
70 | 69 | eqeq2d 2770 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = (2nd ‘𝑧) → (𝑧 = 〈𝑥, 𝑦〉 ↔ 𝑧 = 〈𝑥, (2nd ‘𝑧)〉)) |
71 | 69 | eleq1d 2835 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = (2nd ‘𝑧) → (〈𝑥, 𝑦〉 ∈ 𝐹 ↔ 〈𝑥, (2nd ‘𝑧)〉 ∈ 𝐹)) |
72 | 70, 71 | anbi12d 634 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (2nd ‘𝑧) → ((𝑧 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐹) ↔ (𝑧 = 〈𝑥, (2nd ‘𝑧)〉 ∧ 〈𝑥, (2nd ‘𝑧)〉 ∈ 𝐹))) |
73 | 56, 68, 72 | spcedv 3515 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → ∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐹)) |
74 | | vex 3411 |
. . . . . . . . . . . . . 14
⊢ 𝑥 ∈ V |
75 | 74 | elsnres 5856 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ (𝐹 ↾ {𝑥}) ↔ ∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 〈𝑥, 𝑦〉 ∈ 𝐹)) |
76 | 73, 75 | sylibr 237 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝑧 ∈ (𝐹 ↾ {𝑥})) |
77 | 13 | ad3antrrr 730 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝐹 Fn 𝐴) |
78 | 22 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝑥 ∈ 𝐴) |
79 | | fnressn 6904 |
. . . . . . . . . . . . 13
⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 ↾ {𝑥}) = {〈𝑥, (𝐹‘𝑥)〉}) |
80 | 77, 78, 79 | syl2anc 588 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → (𝐹 ↾ {𝑥}) = {〈𝑥, (𝐹‘𝑥)〉}) |
81 | 76, 80 | eleqtrd 2853 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝑧 ∈ {〈𝑥, (𝐹‘𝑥)〉}) |
82 | | elsni 4532 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ {〈𝑥, (𝐹‘𝑥)〉} → 𝑧 = 〈𝑥, (𝐹‘𝑥)〉) |
83 | 81, 82 | syl 17 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑥 = (1st ‘𝑧)) → 𝑧 = 〈𝑥, (𝐹‘𝑥)〉) |
84 | | simpr 489 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑧 = 〈𝑥, (𝐹‘𝑥)〉) → 𝑧 = 〈𝑥, (𝐹‘𝑥)〉) |
85 | 84 | fveq2d 6655 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑧 = 〈𝑥, (𝐹‘𝑥)〉) → (1st ‘𝑧) = (1st
‘〈𝑥, (𝐹‘𝑥)〉)) |
86 | | fvex 6664 |
. . . . . . . . . . . 12
⊢ (𝐹‘𝑥) ∈ V |
87 | 74, 86 | op1st 7694 |
. . . . . . . . . . 11
⊢
(1st ‘〈𝑥, (𝐹‘𝑥)〉) = 𝑥 |
88 | 85, 87 | eqtr2di 2811 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑧 = 〈𝑥, (𝐹‘𝑥)〉) → 𝑥 = (1st ‘𝑧)) |
89 | 83, 88 | impbida 801 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (𝑥 = (1st ‘𝑧) ↔ 𝑧 = 〈𝑥, (𝐹‘𝑥)〉)) |
90 | 89 | ralrimiva 3111 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → ∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st ‘𝑧) ↔ 𝑧 = 〈𝑥, (𝐹‘𝑥)〉)) |
91 | 51, 55, 90 | rspcedvd 3542 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → ∃𝑣 ∈ (𝐹 ∖ (V × { 0 }))∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st ‘𝑧) ↔ 𝑧 = 𝑣)) |
92 | | reu6 3637 |
. . . . . . 7
⊢
(∃!𝑧 ∈
(𝐹 ∖ (V × {
0
}))𝑥 = (1st
‘𝑧) ↔
∃𝑣 ∈ (𝐹 ∖ (V × { 0
}))∀𝑧 ∈ (𝐹 ∖ (V × { 0 }))(𝑥 = (1st ‘𝑧) ↔ 𝑧 = 𝑣)) |
93 | 91, 92 | sylibr 237 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 supp 0 )) → ∃!𝑧 ∈ (𝐹 ∖ (V × { 0 }))𝑥 = (1st ‘𝑧)) |
94 | 17, 6, 7, 18, 8, 19, 20, 23, 40, 93 | gsummptf1o 19136 |
. . . . 5
⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐹 supp 0 ) ↦ (𝐹‘𝑥))) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st
‘𝑧))))) |
95 | 5, 16, 94 | 3eqtr3d 2802 |
. . . 4
⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st
‘𝑧))))) |
96 | | simpr 489 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) |
97 | 96 | eldifad 3866 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → 𝑧 ∈ 𝐹) |
98 | | funfv1st2nd 7742 |
. . . . . . 7
⊢ ((Fun
𝐹 ∧ 𝑧 ∈ 𝐹) → (𝐹‘(1st ‘𝑧)) = (2nd
‘𝑧)) |
99 | 24, 97, 98 | syl2an2r 685 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → (𝐹‘(1st
‘𝑧)) =
(2nd ‘𝑧)) |
100 | 99 | mpteq2dva 5120 |
. . . . 5
⊢ (𝜑 → (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st
‘𝑧))) = (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦
(2nd ‘𝑧))) |
101 | 100 | oveq2d 7159 |
. . . 4
⊢ (𝜑 → (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦ (𝐹‘(1st
‘𝑧)))) = (𝐺 Σg
(𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦
(2nd ‘𝑧)))) |
102 | 95, 101 | eqtrd 2794 |
. . 3
⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦
(2nd ‘𝑧)))) |
103 | | nfcv 2917 |
. . . 4
⊢
Ⅎ𝑧(1st ‘𝑡) |
104 | | fvex 6664 |
. . . . 5
⊢
(2nd ‘𝑡) ∈ V |
105 | | fvex 6664 |
. . . . 5
⊢
(1st ‘𝑡) ∈ V |
106 | 104, 105 | op2ndd 7697 |
. . . 4
⊢ (𝑧 = 〈(2nd
‘𝑡), (1st
‘𝑡)〉 →
(2nd ‘𝑧) =
(1st ‘𝑡)) |
107 | | resfnfinfin 8822 |
. . . . . 6
⊢ ((𝐹 Fn 𝐴 ∧ (𝐹 supp 0 ) ∈ Fin) →
(𝐹 ↾ (𝐹 supp 0 )) ∈
Fin) |
108 | 13, 19, 107 | syl2anc 588 |
. . . . 5
⊢ (𝜑 → (𝐹 ↾ (𝐹 supp 0 )) ∈
Fin) |
109 | 33, 108 | eqeltrrd 2852 |
. . . 4
⊢ (𝜑 → (𝐹 ∖ (V × { 0 })) ∈
Fin) |
110 | 33 | rneqd 5772 |
. . . . 5
⊢ (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) = ran (𝐹 ∖ (V × { 0
}))) |
111 | | rnresss 5852 |
. . . . . 6
⊢ ran
(𝐹 ↾ (𝐹 supp 0 )) ⊆ ran 𝐹 |
112 | 1 | frnd 6498 |
. . . . . 6
⊢ (𝜑 → ran 𝐹 ⊆ 𝐵) |
113 | 111, 112 | sstrid 3899 |
. . . . 5
⊢ (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) ⊆ 𝐵) |
114 | 110, 113 | eqsstrrd 3927 |
. . . 4
⊢ (𝜑 → ran (𝐹 ∖ (V × { 0 })) ⊆ 𝐵) |
115 | | 2ndrn 7737 |
. . . . 5
⊢ ((Rel
(𝐹 ∖ (V × {
0 }))
∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) →
(2nd ‘𝑧)
∈ ran (𝐹 ∖ (V
× { 0 }))) |
116 | 27, 115 | sylan 584 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) →
(2nd ‘𝑧)
∈ ran (𝐹 ∖ (V
× { 0 }))) |
117 | | relcnv 5932 |
. . . . . . . 8
⊢ Rel ◡𝐹 |
118 | | reldif 5650 |
. . . . . . . 8
⊢ (Rel
◡𝐹 → Rel (◡𝐹 ∖ ({ 0 } ×
V))) |
119 | 117, 118 | mp1i 13 |
. . . . . . 7
⊢ (𝜑 → Rel (◡𝐹 ∖ ({ 0 } ×
V))) |
120 | | 1st2nd 7735 |
. . . . . . 7
⊢ ((Rel
(◡𝐹 ∖ ({ 0 } × V)) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) →
𝑡 = 〈(1st
‘𝑡), (2nd
‘𝑡)〉) |
121 | 119, 120 | sylan 584 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) →
𝑡 = 〈(1st
‘𝑡), (2nd
‘𝑡)〉) |
122 | | cnvdif 5967 |
. . . . . . . . . 10
⊢ ◡(𝐹 ∖ (V × { 0 })) = (◡𝐹 ∖ ◡(V × { 0 })) |
123 | | cnvxp 5979 |
. . . . . . . . . . 11
⊢ ◡(V × { 0 }) = ({ 0 } ×
V) |
124 | 123 | difeq2i 4021 |
. . . . . . . . . 10
⊢ (◡𝐹 ∖ ◡(V × { 0 })) = (◡𝐹 ∖ ({ 0 } ×
V)) |
125 | 122, 124 | eqtri 2782 |
. . . . . . . . 9
⊢ ◡(𝐹 ∖ (V × { 0 })) = (◡𝐹 ∖ ({ 0 } ×
V)) |
126 | 125 | eqimss2i 3947 |
. . . . . . . 8
⊢ (◡𝐹 ∖ ({ 0 } × V)) ⊆
◡(𝐹 ∖ (V × { 0 })) |
127 | 126 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (◡𝐹 ∖ ({ 0 } × V)) ⊆
◡(𝐹 ∖ (V × { 0 }))) |
128 | 127 | sselda 3888 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) →
𝑡 ∈ ◡(𝐹 ∖ (V × { 0 }))) |
129 | 121, 128 | eqeltrrd 2852 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) →
〈(1st ‘𝑡), (2nd ‘𝑡)〉 ∈ ◡(𝐹 ∖ (V × { 0 }))) |
130 | 105, 104 | opelcnv 5714 |
. . . . 5
⊢
(〈(1st ‘𝑡), (2nd ‘𝑡)〉 ∈ ◡(𝐹 ∖ (V × { 0 })) ↔
〈(2nd ‘𝑡), (1st ‘𝑡)〉 ∈ (𝐹 ∖ (V × { 0 }))) |
131 | 129, 130 | sylib 221 |
. . . 4
⊢ ((𝜑 ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) →
〈(2nd ‘𝑡), (1st ‘𝑡)〉 ∈ (𝐹 ∖ (V × { 0 }))) |
132 | 27 | adantr 485 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → Rel (𝐹 ∖ (V × { 0
}))) |
133 | | eqidd 2760 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∪ ◡{𝑧} = ∪
◡{𝑧}) |
134 | | cnvf1olem 7803 |
. . . . . . . . 9
⊢ ((Rel
(𝐹 ∖ (V × {
0 }))
∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧
∪ ◡{𝑧} = ∪
◡{𝑧})) → (∪
◡{𝑧} ∈ ◡(𝐹 ∖ (V × { 0 })) ∧ 𝑧 = ∪
◡{∪ ◡{𝑧}})) |
135 | 134 | simpld 499 |
. . . . . . . 8
⊢ ((Rel
(𝐹 ∖ (V × {
0 }))
∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧
∪ ◡{𝑧} = ∪
◡{𝑧})) → ∪ ◡{𝑧} ∈ ◡(𝐹 ∖ (V × { 0 }))) |
136 | 132, 96, 133, 135 | syl12anc 836 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∪ ◡{𝑧} ∈ ◡(𝐹 ∖ (V × { 0 }))) |
137 | 136, 125 | eleqtrdi 2861 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∪ ◡{𝑧} ∈ (◡𝐹 ∖ ({ 0 } ×
V))) |
138 | | eqeq2 2771 |
. . . . . . . . 9
⊢ (𝑢 = ∪
◡{𝑧} → (𝑡 = 𝑢 ↔ 𝑡 = ∪ ◡{𝑧})) |
139 | 138 | bibi2d 347 |
. . . . . . . 8
⊢ (𝑢 = ∪
◡{𝑧} → ((𝑧 = 〈(2nd ‘𝑡), (1st ‘𝑡)〉 ↔ 𝑡 = 𝑢) ↔ (𝑧 = 〈(2nd ‘𝑡), (1st ‘𝑡)〉 ↔ 𝑡 = ∪
◡{𝑧}))) |
140 | 139 | ralbidv 3124 |
. . . . . . 7
⊢ (𝑢 = ∪
◡{𝑧} → (∀𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))(𝑧 = 〈(2nd
‘𝑡), (1st
‘𝑡)〉 ↔
𝑡 = 𝑢) ↔ ∀𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))(𝑧 = 〈(2nd
‘𝑡), (1st
‘𝑡)〉 ↔
𝑡 = ∪ ◡{𝑧}))) |
141 | 140 | adantl 486 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑢 = ∪
◡{𝑧}) → (∀𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))(𝑧 = 〈(2nd
‘𝑡), (1st
‘𝑡)〉 ↔
𝑡 = 𝑢) ↔ ∀𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))(𝑧 = 〈(2nd
‘𝑡), (1st
‘𝑡)〉 ↔
𝑡 = ∪ ◡{𝑧}))) |
142 | 117, 118 | mp1i 13 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = 〈(2nd
‘𝑡), (1st
‘𝑡)〉) → Rel
(◡𝐹 ∖ ({ 0 } ×
V))) |
143 | | simplr 769 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = 〈(2nd
‘𝑡), (1st
‘𝑡)〉) →
𝑡 ∈ (◡𝐹 ∖ ({ 0 } ×
V))) |
144 | | simpr 489 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = 〈(2nd
‘𝑡), (1st
‘𝑡)〉) →
𝑧 = 〈(2nd
‘𝑡), (1st
‘𝑡)〉) |
145 | | df-rel 5524 |
. . . . . . . . . . . . . 14
⊢ (Rel
(◡𝐹 ∖ ({ 0 } × V)) ↔
(◡𝐹 ∖ ({ 0 } × V)) ⊆ (V
× V)) |
146 | 119, 145 | sylib 221 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (◡𝐹 ∖ ({ 0 } × V)) ⊆ (V
× V)) |
147 | 146 | ad3antrrr 730 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = 〈(2nd
‘𝑡), (1st
‘𝑡)〉) →
(◡𝐹 ∖ ({ 0 } × V)) ⊆ (V
× V)) |
148 | 147, 143 | sseldd 3889 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = 〈(2nd
‘𝑡), (1st
‘𝑡)〉) →
𝑡 ∈ (V ×
V)) |
149 | | 2nd1st 7734 |
. . . . . . . . . . 11
⊢ (𝑡 ∈ (V × V) →
∪ ◡{𝑡} = 〈(2nd
‘𝑡), (1st
‘𝑡)〉) |
150 | 148, 149 | syl 17 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = 〈(2nd
‘𝑡), (1st
‘𝑡)〉) →
∪ ◡{𝑡} = 〈(2nd
‘𝑡), (1st
‘𝑡)〉) |
151 | 144, 150 | eqtr4d 2797 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = 〈(2nd
‘𝑡), (1st
‘𝑡)〉) →
𝑧 = ∪ ◡{𝑡}) |
152 | | cnvf1olem 7803 |
. . . . . . . . . 10
⊢ ((Rel
(◡𝐹 ∖ ({ 0 } × V)) ∧ (𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V)) ∧ 𝑧 = ∪
◡{𝑡})) → (𝑧 ∈ ◡(◡𝐹 ∖ ({ 0 } × V)) ∧ 𝑡 = ∪
◡{𝑧})) |
153 | 152 | simprd 500 |
. . . . . . . . 9
⊢ ((Rel
(◡𝐹 ∖ ({ 0 } × V)) ∧ (𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V)) ∧ 𝑧 = ∪
◡{𝑡})) → 𝑡 = ∪ ◡{𝑧}) |
154 | 142, 143,
151, 153 | syl12anc 836 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑧 = 〈(2nd
‘𝑡), (1st
‘𝑡)〉) →
𝑡 = ∪ ◡{𝑧}) |
155 | 27 | ad3antrrr 730 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪
◡{𝑧}) → Rel (𝐹 ∖ (V × { 0 }))) |
156 | 96 | ad2antrr 726 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪
◡{𝑧}) → 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) |
157 | | simpr 489 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪
◡{𝑧}) → 𝑡 = ∪ ◡{𝑧}) |
158 | | cnvf1olem 7803 |
. . . . . . . . . . 11
⊢ ((Rel
(𝐹 ∖ (V × {
0 }))
∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧
𝑡 = ∪ ◡{𝑧})) → (𝑡 ∈ ◡(𝐹 ∖ (V × { 0 })) ∧ 𝑧 = ∪
◡{𝑡})) |
159 | 158 | simprd 500 |
. . . . . . . . . 10
⊢ ((Rel
(𝐹 ∖ (V × {
0 }))
∧ (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ∧
𝑡 = ∪ ◡{𝑧})) → 𝑧 = ∪ ◡{𝑡}) |
160 | 155, 156,
157, 159 | syl12anc 836 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪
◡{𝑧}) → 𝑧 = ∪ ◡{𝑡}) |
161 | 146 | ad3antrrr 730 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪
◡{𝑧}) → (◡𝐹 ∖ ({ 0 } × V)) ⊆ (V
× V)) |
162 | | simplr 769 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪
◡{𝑧}) → 𝑡 ∈ (◡𝐹 ∖ ({ 0 } ×
V))) |
163 | 161, 162 | sseldd 3889 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪
◡{𝑧}) → 𝑡 ∈ (V × V)) |
164 | 163, 149 | syl 17 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪
◡{𝑧}) → ∪ ◡{𝑡} = 〈(2nd ‘𝑡), (1st ‘𝑡)〉) |
165 | 160, 164 | eqtrd 2794 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) ∧ 𝑡 = ∪
◡{𝑧}) → 𝑧 = 〈(2nd ‘𝑡), (1st ‘𝑡)〉) |
166 | 154, 165 | impbida 801 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) ∧ 𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))) →
(𝑧 = 〈(2nd
‘𝑡), (1st
‘𝑡)〉 ↔
𝑡 = ∪ ◡{𝑧})) |
167 | 166 | ralrimiva 3111 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∀𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))(𝑧 = 〈(2nd
‘𝑡), (1st
‘𝑡)〉 ↔
𝑡 = ∪ ◡{𝑧})) |
168 | 137, 141,
167 | rspcedvd 3542 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∃𝑢 ∈ (◡𝐹 ∖ ({ 0 } × V))∀𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))(𝑧 = 〈(2nd
‘𝑡), (1st
‘𝑡)〉 ↔
𝑡 = 𝑢)) |
169 | | reu6 3637 |
. . . . 5
⊢
(∃!𝑡 ∈
(◡𝐹 ∖ ({ 0 } × V))𝑧 = 〈(2nd
‘𝑡), (1st
‘𝑡)〉 ↔
∃𝑢 ∈ (◡𝐹 ∖ ({ 0 } × V))∀𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))(𝑧 = 〈(2nd
‘𝑡), (1st
‘𝑡)〉 ↔
𝑡 = 𝑢)) |
170 | 168, 169 | sylibr 237 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐹 ∖ (V × { 0 }))) → ∃!𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V))𝑧 = 〈(2nd
‘𝑡), (1st
‘𝑡)〉) |
171 | 103, 6, 7, 106, 8, 109, 114, 116, 131, 170 | gsummptf1o 19136 |
. . 3
⊢ (𝜑 → (𝐺 Σg (𝑧 ∈ (𝐹 ∖ (V × { 0 })) ↦
(2nd ‘𝑧)))
= (𝐺
Σg (𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V)) ↦
(1st ‘𝑡)))) |
172 | | fveq2 6651 |
. . . . . 6
⊢ (𝑡 = 𝑧 → (1st ‘𝑡) = (1st ‘𝑧)) |
173 | 172 | cbvmptv 5128 |
. . . . 5
⊢ (𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V)) ↦
(1st ‘𝑡))
= (𝑧 ∈ (◡𝐹 ∖ ({ 0 } × V)) ↦
(1st ‘𝑧)) |
174 | 33 | cnveqd 5708 |
. . . . . . 7
⊢ (𝜑 → ◡(𝐹 ↾ (𝐹 supp 0 )) = ◡(𝐹 ∖ (V × { 0 }))) |
175 | 174, 125 | eqtr2di 2811 |
. . . . . 6
⊢ (𝜑 → (◡𝐹 ∖ ({ 0 } × V)) = ◡(𝐹 ↾ (𝐹 supp 0 ))) |
176 | 175 | mpteq1d 5114 |
. . . . 5
⊢ (𝜑 → (𝑧 ∈ (◡𝐹 ∖ ({ 0 } × V)) ↦
(1st ‘𝑧))
= (𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st
‘𝑧))) |
177 | 173, 176 | syl5eq 2806 |
. . . 4
⊢ (𝜑 → (𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V)) ↦
(1st ‘𝑡))
= (𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st
‘𝑧))) |
178 | 177 | oveq2d 7159 |
. . 3
⊢ (𝜑 → (𝐺 Σg (𝑡 ∈ (◡𝐹 ∖ ({ 0 } × V)) ↦
(1st ‘𝑡)))
= (𝐺
Σg (𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st
‘𝑧)))) |
179 | 102, 171,
178 | 3eqtrd 2798 |
. 2
⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st
‘𝑧)))) |
180 | | nfcv 2917 |
. . 3
⊢
Ⅎ𝑦(1st ‘𝑧) |
181 | | nfv 1916 |
. . 3
⊢
Ⅎ𝑥𝜑 |
182 | | vex 3411 |
. . . 4
⊢ 𝑦 ∈ V |
183 | 74, 182 | op1std 7696 |
. . 3
⊢ (𝑧 = 〈𝑥, 𝑦〉 → (1st ‘𝑧) = 𝑥) |
184 | | relcnv 5932 |
. . . 4
⊢ Rel ◡(𝐹 ↾ (𝐹 supp 0 )) |
185 | 184 | a1i 11 |
. . 3
⊢ (𝜑 → Rel ◡(𝐹 ↾ (𝐹 supp 0 ))) |
186 | | cnvfi 8824 |
. . . 4
⊢ ((𝐹 ↾ (𝐹 supp 0 )) ∈ Fin → ◡(𝐹 ↾ (𝐹 supp 0 )) ∈
Fin) |
187 | 108, 186 | syl 17 |
. . 3
⊢ (𝜑 → ◡(𝐹 ↾ (𝐹 supp 0 )) ∈
Fin) |
188 | 112 | adantr 485 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) → ran 𝐹 ⊆ 𝐵) |
189 | 184 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) → Rel ◡(𝐹 ↾ (𝐹 supp 0 ))) |
190 | | simpr 489 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) → 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) |
191 | | 1stdm 7736 |
. . . . . . 7
⊢ ((Rel
◡(𝐹 ↾ (𝐹 supp 0 )) ∧ 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) → (1st
‘𝑧) ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) |
192 | 189, 190,
191 | syl2anc 588 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) → (1st
‘𝑧) ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) |
193 | | df-rn 5528 |
. . . . . 6
⊢ ran
(𝐹 ↾ (𝐹 supp 0 )) = dom ◡(𝐹 ↾ (𝐹 supp 0 )) |
194 | 192, 193 | eleqtrrdi 2862 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) → (1st
‘𝑧) ∈ ran (𝐹 ↾ (𝐹 supp 0 ))) |
195 | 111, 194 | sseldi 3886 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) → (1st
‘𝑧) ∈ ran 𝐹) |
196 | 188, 195 | sseldd 3889 |
. . 3
⊢ ((𝜑 ∧ 𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 ))) → (1st
‘𝑧) ∈ 𝐵) |
197 | 180, 181,
6, 183, 185, 187, 8, 196 | gsummpt2d 30820 |
. 2
⊢ (𝜑 → (𝐺 Σg (𝑧 ∈ ◡(𝐹 ↾ (𝐹 supp 0 )) ↦ (1st
‘𝑧))) = (𝐺 Σg
(𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 )) ↦ (𝐺 Σg
(𝑦 ∈ (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥))))) |
198 | | df-ima 5530 |
. . . . . . 7
⊢ (𝐹 “ (𝐹 supp 0 )) = ran (𝐹 ↾ (𝐹 supp 0 )) |
199 | | supppreima 30534 |
. . . . . . . . 9
⊢ ((Fun
𝐹 ∧ 𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (◡𝐹 “ (ran 𝐹 ∖ { 0 }))) |
200 | 24, 12, 31, 199 | syl3anc 1369 |
. . . . . . . 8
⊢ (𝜑 → (𝐹 supp 0 ) = (◡𝐹 “ (ran 𝐹 ∖ { 0 }))) |
201 | 200 | imaeq2d 5894 |
. . . . . . 7
⊢ (𝜑 → (𝐹 “ (𝐹 supp 0 )) = (𝐹 “ (◡𝐹 “ (ran 𝐹 ∖ { 0 })))) |
202 | 198, 201 | syl5eqr 2808 |
. . . . . 6
⊢ (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) = (𝐹 “ (◡𝐹 “ (ran 𝐹 ∖ { 0 })))) |
203 | | funimacnv 6409 |
. . . . . . 7
⊢ (Fun
𝐹 → (𝐹 “ (◡𝐹 “ (ran 𝐹 ∖ { 0 }))) = ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹)) |
204 | 24, 203 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝐹 “ (◡𝐹 “ (ran 𝐹 ∖ { 0 }))) = ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹)) |
205 | | difssd 4034 |
. . . . . . 7
⊢ (𝜑 → (ran 𝐹 ∖ { 0 }) ⊆ ran 𝐹) |
206 | | df-ss 3871 |
. . . . . . 7
⊢ ((ran
𝐹 ∖ { 0 }) ⊆
ran 𝐹 ↔ ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹) = (ran 𝐹 ∖ { 0 })) |
207 | 205, 206 | sylib 221 |
. . . . . 6
⊢ (𝜑 → ((ran 𝐹 ∖ { 0 }) ∩ ran 𝐹) = (ran 𝐹 ∖ { 0 })) |
208 | 202, 204,
207 | 3eqtrd 2798 |
. . . . 5
⊢ (𝜑 → ran (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 })) |
209 | 193, 208 | syl5eqr 2808 |
. . . 4
⊢ (𝜑 → dom ◡(𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 })) |
210 | 8 | cmnmndd 18981 |
. . . . . . 7
⊢ (𝜑 → 𝐺 ∈ Mnd) |
211 | 210 | adantr 485 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → 𝐺 ∈ Mnd) |
212 | 108 | adantr 485 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 ↾ (𝐹 supp 0 )) ∈
Fin) |
213 | | imafi2 30555 |
. . . . . . 7
⊢ (◡(𝐹 ↾ (𝐹 supp 0 )) ∈ Fin →
(◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ∈ Fin) |
214 | 212, 186,
213 | 3syl 18 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ∈ Fin) |
215 | 193, 113 | eqsstrrid 3937 |
. . . . . . 7
⊢ (𝜑 → dom ◡(𝐹 ↾ (𝐹 supp 0 )) ⊆ 𝐵) |
216 | 215 | sselda 3888 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → 𝑥 ∈ 𝐵) |
217 | | gsumhashmul.x |
. . . . . . 7
⊢ · =
(.g‘𝐺) |
218 | 6, 217 | gsumconst 19107 |
. . . . . 6
⊢ ((𝐺 ∈ Mnd ∧ (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ∈ Fin ∧ 𝑥 ∈ 𝐵) → (𝐺 Σg (𝑦 ∈ (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)) = ((♯‘(◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})) · 𝑥)) |
219 | 211, 214,
216, 218 | syl3anc 1369 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (𝐺 Σg
(𝑦 ∈ (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)) = ((♯‘(◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})) · 𝑥)) |
220 | | cnvresima 6052 |
. . . . . . . 8
⊢ (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) = ((◡𝐹 “ {𝑥}) ∩ (𝐹 supp 0 )) |
221 | 209 | eleq2d 2836 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 )) ↔ 𝑥 ∈ (ran 𝐹 ∖ { 0 }))) |
222 | 221 | biimpa 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → 𝑥 ∈ (ran 𝐹 ∖ { 0 })) |
223 | 222 | snssd 4692 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → {𝑥} ⊆ (ran 𝐹 ∖ { 0 })) |
224 | | sspreima 30489 |
. . . . . . . . . . 11
⊢ ((Fun
𝐹 ∧ {𝑥} ⊆ (ran 𝐹 ∖ { 0 })) → (◡𝐹 “ {𝑥}) ⊆ (◡𝐹 “ (ran 𝐹 ∖ { 0 }))) |
225 | 24, 223, 224 | syl2an2r 685 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (◡𝐹 “ {𝑥}) ⊆ (◡𝐹 “ (ran 𝐹 ∖ { 0 }))) |
226 | 200 | adantr 485 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (𝐹 supp 0 ) = (◡𝐹 “ (ran 𝐹 ∖ { 0 }))) |
227 | 225, 226 | sseqtrrd 3929 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (◡𝐹 “ {𝑥}) ⊆ (𝐹 supp 0 )) |
228 | | df-ss 3871 |
. . . . . . . . 9
⊢ ((◡𝐹 “ {𝑥}) ⊆ (𝐹 supp 0 ) ↔ ((◡𝐹 “ {𝑥}) ∩ (𝐹 supp 0 )) = (◡𝐹 “ {𝑥})) |
229 | 227, 228 | sylib 221 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → ((◡𝐹 “ {𝑥}) ∩ (𝐹 supp 0 )) = (◡𝐹 “ {𝑥})) |
230 | 220, 229 | syl5req 2807 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (◡𝐹 “ {𝑥}) = (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})) |
231 | 230 | fveq2d 6655 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) →
(♯‘(◡𝐹 “ {𝑥})) = (♯‘(◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}))) |
232 | 231 | oveq1d 7158 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) →
((♯‘(◡𝐹 “ {𝑥})) · 𝑥) = ((♯‘(◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥})) · 𝑥)) |
233 | 219, 232 | eqtr4d 2797 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 ))) → (𝐺 Σg
(𝑦 ∈ (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)) = ((♯‘(◡𝐹 “ {𝑥})) · 𝑥)) |
234 | 209, 233 | mpteq12dva 5109 |
. . 3
⊢ (𝜑 → (𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 )) ↦ (𝐺 Σg
(𝑦 ∈ (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥))) = (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦
((♯‘(◡𝐹 “ {𝑥})) · 𝑥))) |
235 | 234 | oveq2d 7159 |
. 2
⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ dom ◡(𝐹 ↾ (𝐹 supp 0 )) ↦ (𝐺 Σg
(𝑦 ∈ (◡(𝐹 ↾ (𝐹 supp 0 )) “ {𝑥}) ↦ 𝑥)))) = (𝐺 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦
((♯‘(◡𝐹 “ {𝑥})) · 𝑥)))) |
236 | 179, 197,
235 | 3eqtrd 2798 |
1
⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑥 ∈ (ran 𝐹 ∖ { 0 }) ↦
((♯‘(◡𝐹 “ {𝑥})) · 𝑥)))) |